You probably have heard about alleged vote fraud in the Russian legislative election 2011. You might even have seen videos about cases of vote rigging. The ruling party dismisses allegations, because even if some of the cases may have been proven, there are too few of these cases to jeopardize the legitimacy of the whole election. Some of the opposition disagrees with this and is taking it to the streets. There were over 95000 polling stations in the election and of course election observers were not present in all of them. So few Russian citizens wondered if there would be some statistical support to the claims of systematic vote rigging. I have gathered information from three different Russian blog-posts that present such statistics. There are thousands of different reasons that can slightly alter the vote turnout at any given polling station. Some of these reasons affect the turnout positively, some of them negatively, but since the reasons are mainly random it is extremely unlikely that the turnout of a certain polling station differs a great deal from the average turnout of the elections. Usually when you plot the polling station by their turnout percentage you get a Normal distribution . Here's a picture:

In the above picture the X-axis shows vote turnout and the Y-axis shows amount of polling stations. From left to right the four curves represent Mexican legislative election 2009, second round of Polish presidential election in 2010, Bulgarian legislative election in 2009 and Swedish legislative election in 2010. Tendency towards a normal distribution is obvious. Now let's take a look if Russian elections in 2011 follow the same logic:

You can see the normal distribution all the way to the 55% turnout. Then something strange happens. For some reason turnout significantly larger than the mean is far more likely than turnout significantly lower than the mean. This cannot be explained by existence of very small polling stations since the results are weighted. So maybe it's just one of those Russian things like excessive alcohol consumption. Let's take a look at the previous four election in Russia:

The pale brown curve is for the 2003 legislative election, the dark brown curve is for the 2007 legislative election, the light blue curve is for the 2004 presidential election and the dark blue curve is for the 2008 presidential election. Once again we observe normal distribution until the 55% turnout mark. But take a look at those spikes at 70%, 75%, 80%, 85%, 90% and 98%! There are two ways to explain this. First explanation is that for some unknown reason voters of large amount of polling stations coordinate their actions by setting a target turnout. The second explanation is that there is systematic vote fraud at some of the polling station with specific target amount of votes. Take for example the 98% spikes in the curves. Imagine that you are committing a vote fraud. You don't want to get caught and 100% turnout is easily proven false as soon as one of the citizens of the area tells that he didn't vote since he was out of town. So, since you live in Russia and not North Korea, you make the turnout 98%. Take for example the 2011 legislative election in Chechnya: the turnout is 93.31% and 99.48% of votes go to United Russia party. One could use the argument that the even percentage spikes are there because of very small polling stations. In a polling station with 20 voters all of the possible turnout outcomes end with 5 or 0. But there are not that many small polling stations, and even if there were, you would expect the similar spikes at polling stations with low turnout.

So, now that we concluded that there is something fishy about Russian election, the next question is who benefits from this? The opposition blames the United Russia party, which holds the majority of lower and higher house parliament, as well as the presidential seat. United Russia denies vote fraud. Accusing opposition of vote rigging would take away the legitimacy of the election and if you're the one in power you don't want to do that. One might ask why would United Russia, Putin or Medvedev resort to vote fraud despite the fact that they are truly popular in Russia and would get a majority or close to a majority of parliament seats/votes even without the fraud. That is a fair question, but politics is a game that has wide array of possible outcomes, not just win or lose. It is a completely different thing to do what you want when you can claim a mandate of 80% of the people instead of just half. But is there evidence that the United Russia is benefiting from vote fraud? Imagine that you are going to a polling station and that you already know who are you going to vote. At the door to the polling station you find out that the turnout there is already over 40%/60%/80%. Does that influence your vote? Probably not. Let's take a look the next picture.

Picture on the left is about Russian legislative election 2007, the second picture is from Moscow city parliament election in 2009. Every dot represents the amount of votes that a party had at a polling station. The X-axis is for turnout of the polling station and the Y-axis is for the percentage of votes a party had at that station. Different colors represent different parties. Take a guess what party the purple dots represents. That's right, United Russia. Now lets take a look at the 2011 election results from Moscow:

The Y-axis is for the amount of polling stations, the X-axis is the vote percentage. We see curves for five different parties, four of them are close to the normal distribution curves and one of them belongs to the United Russia. You see two spikes: one at 30% which is what surveys predicted for United Russia and one at 50-55% which is what was probably United Russia's own target. Now the vote result in the city of Magnitogorsk:

This is a curve of United Russia votes in Magnitogorsk, a South Russia city with a population of 400000. I guess the people there are really divided or something. And then there's the last picture about the correlation between the vote turnout and the vote percentage of United Russia:

In conclusion, I would say that the vote fraud was definitely there, but it's to early to tell how much it affected the result. United Russia would probably have been the biggest party even without the fraud. But that's not the point. The point is that a rigged election does not give legitimacy to the new Duma. Even without the vote fraud there's still the issue of not allowing all opposition parties to take part in the elections and the lack of freedom of press. I, as a citizen of Russia voted in these elections at a Russian embassy in Helsinki. Yabloko party received 25% of the votes here, 22 points more than in Russian regions while United Russia got less than 20%. Is it because people with more "western" views are more likely to migrate from Russia or because of the free press here? That is a topic for a whole other discussion.